Relaxation of the incompressible porous media equation

TL;DR

This paper explicitly calculates the relaxation of the incompressible porous media equation, enabling the construction of weak solutions to related interface problems without complex configurations, thus advancing understanding of their mathematical structure.

Contribution

It provides an explicit relaxation of the IPM equation, bypassing T4 configurations, and applies this to construct weak solutions for the Muskat problem.

Findings

Explicit relaxation of IPM calculated

Weak solutions for Muskat problem constructed

Simplifies previous convex integration approaches

Abstract

It was shown recently by Cordoba, Faraco and Gancedo that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework, uses ideas from the theory of laminates, in particular T4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T4 configurations. We then use this to construct weak solutions to the unstable interface problem (the Muskat problem), as a byproduct shedding new light on the gradient flow approach introduced by Otto.